• Journal of Elementary Mathematics Education in Korea
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• v.21 no.1
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• pp.115-134
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• 2017

It is necessary to understand the characteristics of each type of division problems in other to help students develop a rich understanding when they learn each type of division problems. This study focuses on a specific type of division problems; a quotitive division as finding a scale factor in enlargement context. First, this study investigated via survey how 4th-6th graders and preservice and inservice elementary teachers solved a quotitive division relating to scaling problem. And semi-structured interviews with preservice and inservice elementary teachers were conducted to explore what knowledge they brought when they tried to solve enlargement quotitive division problems. Most of participants solved the given quotitive division problem in the same way. Only a few preservice and inservice teachers interpreted it as a proportion problem and solved in a different way. From the interviews, it was found that different conceptions of context and decontextualization, and different conceptions of times (as repeated addition or as a multiplicative operator) were connected to different solutions. Finally, three issues relating to teaching enlargement quotitive division were discussed; visual representation of two solutions, conceptions connected each solution, and integrating quotitive division and proportion in math textbooks.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.3
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• pp.395-411
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• 2013

There are two concepts of division: measurement division and partitive or fair-sharing division. Students are expected to understand comprehensively division algorithm in both contexts. Contents of textbooks and teachers' guidebooks should be suitable for helping students develop comprehensive understanding of algorithm for whole-number division in both contexts. The results of the analysis of textbooks and teachers' guidebooks shows that they fail to connect two division contexts with division algorithm comprehensively. Their expedient and improper use of two division contexts would keep students from developing comprehensive understanding of algorithm for whole-number division. Based on the results of analysis, some ways of improving textbooks and teachers' guidebooks are suggested.

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.2
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• pp.319-339
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• 2014

Recently, the discussion about division and partition of fraction increases in Korea's national curriculum documents. There are varieties of assertions arranging from the opinion that both interpretations are unintelligible to the opinion that both interpretations are intelligible. In this paper, we investigated a possibility that division and partition interpretation of fraction become valid. As a result, it is appeared that division and partition interpretation of fraction can be defined reasonably through expansion of interpretation of natural number. Besides, division and partition interpretation of fraction can be work in activity, such as constructing equation from sentence problem, or such as proving algorithm of fraction division.

• Education of Primary School Mathematics
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• v.14 no.3
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• pp.317-328
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• 2011

We analyzed errors committed by Korean prospective elementary teachers in finding and interpreting quotient and remainder within measurement division of fractions. 65 prospective elementary teachers were participated in this study. They solved a word problem about measurement division of fractions. We analyzed solutions of all participants, and interviewed 5 participants of them. The results reveal many of these prospective teachers could not tell what fractional part of division result means. Thses results suggest that teacher preparation program should emphasize interpreting calculation results within given situations.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.4
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• pp.521-539
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• 2016

The structures of partitive and quotitive division of fractions are dealt with differently, and this led to using partitive division context for helping develop invert-multiply algorithm and quotitive division for common denominator algorithm. This approach is unlikely to provide children with an opportunity to develop an understanding of common structure involved in solving different types of division. In this study, I propose two approaches, measurement approach and isomorphism approach, to develop a unifying understanding of fraction division. From each of two approaches of solving quotitive division based on proportional reasoning, I discuss an idea of constructing a measure space, unit of which is a quantity of divisor, and another idea of constructing an isomorphic relationship between the measure spaces of dividend and divisor. These ideas support invert-multiply algorithm for quotitive as well as partitive division and bring proportional reasoning into the context of fraction division. I also discuss some curriculum issues regarding fraction division and proportion in order to promote the proposed unifying understanding of partitive and quotitive division of fractions.

• School Mathematics
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• v.6 no.3
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• pp.235-249
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• 2004

The purpose of this study was to analyze the error patterns and sentence types in word problems with respect to 1$\frac{3}{4}$$\div$$\frac{1}{2}$ which were made by the pre-service elementary teachers, and to suggest the clues to the education in pre-service. Korean elementary teachers in pre-service misunderstood 'divide with $\frac{1}{2}$' to 'divide to 2' by the Korean linguistic structure. And they showed a new error type of 1$\frac{3}{4}$$\times$2 by the result of calculation. Although they are familiar to 'inclusive algorithm' they are not good at dealing with the fractional divisor. And they are very poor at the 'decision the unit proportion' and the 'inverse of multiplication'. So, it is necessary to teach the meaning of the fractional division as 'decision the unit proportion' and 'inverse of multiplication' and to give several examples with respect to the actual situation and context.

• Journal of Educational Research in Mathematics
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• v.23 no.1
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• pp.17-35
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• 2013

The purpose of this study was to analyze the extendibility of division algorithm and Euclid algorithm for integers to algorithms for rational numbers based on word problems of fraction division. This study serviced to upgrade professional development of elementary and secondary mathematics teachers. In this paper, fractions were used as expressions of rational numbers, and they also represent rational numbers. According to discrete context and continuous context, and measurement division and partition division etc, divisibility was classified into two types; one is an abstract algebraic point of view and the other is a generalizing view which preserves division algorithms for integers. In the second view, we raised some contextual problems that can be used in school mathematics and then we discussed division algorithm, the greatest common divisor and the least common multiple, and Euclid algorithm for fractions.

• Education of Primary School Mathematics
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• v.22 no.2
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• pp.113-127
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• 2019

The purpose of this study is to justify the fraction division algorithm in elementary mathematics by applying the definition of natural number division to fraction division. First, we studied the contents which need to be taken into consideration in teaching fraction division in elementary mathematics and suggested the criteria. Based on this research, we examined whether the previous methods which are used to derive the standard algorithm are appropriate for the course of introducing the fraction division. Next, we defined division in fraction and suggested the unit-circle partition model and the square partition model which can visualize the definition. Finally, we confirmed that the standard algorithm of fraction division in both partition and measurement is naturally derived through these models.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.4
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• pp.385-403
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• 2018

Fraction division can be categorized as partitive division, measurement division, and the inverse of a Cartesian product. In the contexts of quotitive division and the inverse of a Cartesian product, the multiply-by-the-reciprocal algorithm is drawn well out. In this study, I analyze the potential and significance of the method of using $1{\div}$(divisor) as an alternative way of developing the multiply-by-the-reciprocal algorithm in the context of quotitive division. The method of using $1{\div}$(divisor) in quotitive division has the following advantages. First, by this method we can draw the multiply-by-the-reciprocal algorithm keeping connection with the context of quotitive division. Second, as in other contexts, this method focuses on the multiplicative relationship between the divisor and 1. Third, as in other contexts, this method investigates the multiplicative relationship between the divisor and 1 by two kinds of reasoning that use either ${\frac{1}{the\;denominator\;of\;the\;divisor}}$ or the numerator of the divisor as a stepping stone. These advantages indicates the potential of this method in understanding the multiply-by-the-reciprocal algorithm as the common structure of fraction division. This method is based on the dual meaning of a fraction as a quantity and the composition of times which the current elementary mathematics textbook does not focus on. It is necessary to pay attention to how to form this basis when developing teaching materials for fraction division.

• School Mathematics
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• v.12 no.4
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• pp.585-604
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• 2010

This study aims to investigate understanding of meanings of division, quotitive division and partitive division, by the third graders and preservice elementary teachers. To do this, we analysed and compared mathematics textbooks according to 9 mathematics curricula, gathered information about their understanding by questionnaire method targeting 5 third graders and 36 preservice elementary teachers, and analysed their responses in relation to recognition of division-based situations, solution using visual representations, and awareness of quotitive division and partitive division. In Korea, meanings of division have been taught in grade 2 or 3 in various ways according to curricula. In particular, the mathematics textbook of present curriculum shows a couple of radical changes in relation to introduction of division. We raised the necessity of reexamination of these changes, based on our results from questionnaire analysis that show lack of understanding about two meanings of division by the preservice elementary teachers as well as the third graders. And we also induced several didactical implications for teaching meanings of division.